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[gentle music]

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- David Ebert:
Well, good afternoon, everyone.

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Welcome to the beautiful
Green Lake Conference Center

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for the Wisconsin Mathematics
Council annual conference.

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Today, we are partnering

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with Wisconsin PBS
and Badger Talks

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to share
Jordan Ellenberg's message

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with teachers across the state.

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My name is Dave Ebert,

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and I'm excited to introduce
today's keynote speaker.

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Jordan Ellenberg
is a number theorist

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and professor of mathematics

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at the
University of Wisconsin-Madison.

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He's the author
of numerous books,

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including the bestseller

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<i>How Not to Be Wrong: The Power</i>
<i>of Mathematical Thinking,</i>

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and the upcoming
<i>Don't Be Too Sure.</i>

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His popular articles
about mathematics

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have appeared
in <i>The New York Times,</i>

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<i>The Wall Street Journal,</i>
<i>The Washington Post,</i>

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<i>Wired,</i> and <i>Slate.</i>

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In April of 2026,

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he was elected to the American
Academy of Arts and Sciences.

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Please join me in giving
a warm Wisconsin welcome

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to Jordan Ellenberg.

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[audience applauds]

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- Jordan Ellenberg: Okay,
hi, everybody.

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Thank you guys so much for
coming and so much for teaching.

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I wanna talk about uncertainty
today.

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It might not be obvious
at first,

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like, that this talk
was gonna be about math.

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But bear with me.

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We're gonna get there.

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But let me start.

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There is actually
a mathematician on this slide,

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W.K. Clifford.

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So, let me start with what
I think is the basic dispute

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that I wanna talk about:

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the dispute between certainty
and uncertainty,

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between Clifford, who is both a
philosopher and a mathematician,

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writing, "It is wrong, always,
everywhere, and for everyone

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to believe anything
upon insufficient evidence."

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So, really this brief

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for, like, knowing
exactly what is true

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and not settling for any less.

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Versus the philosopher
and psychologist William James,

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writing in response to Clifford,

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saying,
"Not a victory is gained,

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"not a deed of faithfulness
or courage is done

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except upon a maybe."

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So, he's kind of
one of the great apostles

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of maybe-ism, if you like.

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And I want to sort of try
to say something

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about maybe
how we can bridge the gap

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between these two great thinkers
of the 19th century

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and do a little math as well.

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So, we'll start here.

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This is, of course, a platypus,

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"an amusing example
of the anomalies

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by which Nature sports with
our schemes of classification."

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This is another quote
from a famous person,

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but probably not one
of his most famous quotes,

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unless somebody knows it.

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This is, of course,
Thomas Jefferson,

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who, writing in a letter

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to the Philadelphia doctor
John Manners in 1812.

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So, let me just emphasize

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it used to be
that if you were confused

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about whether the platypus
was a mammal or a bird,

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you could just write a former
president of the United States

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and he would, like, write you
back with his opinions.

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I mean, really,
when you read these documents,

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you really learn
that our country

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was founded by weird nerds,
which I think is--

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like Thomas Jefferson,

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which I think
is very interesting.

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But this was a very hot dispute

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in the biology
of the time, right?

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Because the platypus lays eggs,

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and it has a bill and a cloaca
as if it were a bird,

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but it also has warm blood
and is kind of furry

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and makes milk for its young
like a mammal.

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So, this was something
that people argued about a lot,

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not just former presidents.

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In fact, the British naturalist
George Shaw

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was sure it was actually a hoax.

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He was like,
"Somebody has to--

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"This thing that was mailed to
us from Australia, this carcass,

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"it has to have been, like,
stitched together

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from different animals."

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But he couldn't find
the stitches.

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And when we think...

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You know, what kind of thing
is this?

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Here, we're gonna come back
to this quote a little later.

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But just to sort of show you

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that this was a very
long-standing argument.

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Here's Shakespeare,
already writing...

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Falstaff saying of the maid
in this hotel,

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"She's neither fish nor flesh."

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This might be an expression
you know, right?

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Sort of like,
"What kind of beast is she?"

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In this case,
it's not a mammal or a bird.

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It's like, a land animal
or a sea animal.

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So, this kind of question
was already in people's mind.

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And if you know one name

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that's associated with the
classification of animals,

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it's certainly this guy,
Carl Linnaeus.

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This is his first big book,
<i>The Sexual System of Plants,</i>

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where he classified all plants

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exactly by the
sort of organization

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of their sexual organs.

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He, also-- I couldn't fit them
all in this slide,

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but there's I think 24 or 25
different classifications

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given by letters,
which gave, like, you know,

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rather, like, racy names

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like seven husbands
and one wife,

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like, depending on how

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the pistils and stamens
were organized.

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But he was, in some sense,
the great promoter

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of the idea
of strict classification.

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And, in fact, I'm gonna
read you this quote.

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I say,
"The first step in wisdom--"

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And that's a quote
from Linnaeus.

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He writes in this book,

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"The first step in wisdom is
to know the things themselves.

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"This notion consists in having
a true idea of the objects.

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"Objects are distinguished
and known

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"by classifying them
methodically

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"and giving them
appropriate names.

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"Therefore, classification
and name-giving

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will be the foundation
of our science."

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So, again, we know from biology,
right?

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These Latin two-part names that
we use for species even today,

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this was Linnaeus's idea,

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that we're gonna sort of give
everything its exact proper name

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that cleaves it apart
from all other species.

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And for him,
this was not just

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sort of a part
of what science should do.

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This is, like, really
what science is.

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This is what it is
to know something.

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So, this is the exact idea

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to which the platypus does a
certain kind of violence, right?

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We're sort of not happy
if we're Linnaeus

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and we encounter this animal.

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That being said, I mean,
we like plants, we like animals.

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But what we as people are
most interested in is people.

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And so,
we're just as interested--

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and actually always have been

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since the time
of the four humors--

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as interested
in classifying people--

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more interested than we are in
classifying animals and plants.

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I would say, in fact,
I would say as long

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as there have been people,
there have been kinds of people.

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I like to always have, like,

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a little Wisconsin content
in my talks,

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especially when I'm talking
here in Wisconsin.

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So, here's another important
source document.

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And I'm curious if anybody knows
who made this.

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Where we have this box
with neutrality in the center,

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chaos and law
on the horizontal axis,

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and good and evil--

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Okay, you know in the middle.
- Attendee 1: Gary Gygax.

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- This is indeed a Gary Gygax--

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Wait, does everybody in the room
know who E. Gary Gygax is?

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Oh, my God, okay.

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E. Gary Gygax is, of course, the
creator of <i>Dungeons and Dragons,</i>

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one of the most important
cultural products

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ever to come out
of the great state of Wisconsin.

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He lived in Lake Geneva.

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And what you're looking at here
is an article of his from 1973,

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which was the introduction--
the first-ever instance

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of Gary Gygax's
famous classification

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of all <i>D and D</i> characters
into, like, four moral types:

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chaotic good, chaotic evil,
lawful good, and lawful evil.

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So, this idea
of types of people--

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And actually, I'm gonna quote
from his article.

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He writes in 1973,
"Few humans are chaotic,

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and very few
are chaotic and evil."

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Those were the days, right?

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Okay.
[audience laughs]

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So, this kind of classification,

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especially what you might call
a dimensional classification,

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where there's the dimension
of chaos versus law

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and the dimension
of good versus evil

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is incredibly popular.

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Here, I'm just gonna, I have a
little slide with just, like,

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a few of the many
classifications

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that people tend to use.

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There are
the five love languages.

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On the left is what's called
the Enneagram.

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And then, of course,
the horoscope,

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which people use all the time

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as a kind of folk classification
of people into types.

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But maybe
one of the most famous ones

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that you hear about all the time

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was made by these two people,
who I wonder--

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this is another hard one--

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if anybody can identify
who these two women are.

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The same guy?

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I want somebody else to answer.

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Did you look at my notes
before I came? [laughs]

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Okay, who are these two people?
- Attendee 1: Myers and Briggs.

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- These are Myers and Briggs of
the Myers-Briggs Type Inventory,

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or MBTI,

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which even if you don't know
the initials MBTI,

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it's this thing here.

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They were twice as uptight
about classification

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as E. Gary Gygax.

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Instead of two dimensions,
they had four.

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And so, now we're gonna do
a tiny bit of arithmetic.

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There's gonna be more math later
in this talk.

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So, the total number of types,

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there's two things
that each characteristic can be.

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You can be an "E" or an "I,"
an "N" or an "S,"

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a "T" or an "F," a "P" or a "J."

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What those mean is not important
for this talk.

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And that gives you
16 different types of people.

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And I put ENTP first
'cause that's what I am.

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I don't know if I have any
other ENTPs in the room today.

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And this classification
is incredibly popular.

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You see it everywhere.

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There's actually,
like, a K-pop album

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that sold 2.5 million copies
in Korea called <i>ISTJ,</i>

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which is about the sad story
of an ISTJ

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who falls in love with an ENFP.

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[audience murmurs]

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I got to read you this quote,
because...

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So, here on the left,
you see, like, a parenting book,

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which is entirely MBTI-oriented,

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like how to parent each of the
16 types of children.

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And I just want to read you
a quote from this book,

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because I think
it really speaks to--

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I would not go so far as to say,

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like, the kind of anxiety that
this classification speaks to.

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Ready?

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So, here's what it says.

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This book starts with the story
of Lisa and Barry.

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It's from the '80s.

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That's why the people are called
Lisa and Barry.

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Who were very stressed-out
parents.

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It says, "They were often
at a loss as to how to parent.

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"They felt mystified.

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"They felt
they were out of their league.

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Lisa and Barry are hardly
alone," the book says.

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And then they say the following.

246
00:09:58,298 --> 00:09:59,533
They say,

247
00:09:59,600 --> 00:10:04,104
"But what if we did know
from early in our child's life

248
00:10:04,171 --> 00:10:06,073
"who she really was?

249
00:10:06,440 --> 00:10:09,243
"What if we could figure out
by watching her interactions,

250
00:10:09,309 --> 00:10:12,880
"her play or word choices,
her decision-making style,

251
00:10:12,946 --> 00:10:15,115
"what kind of person she is?

252
00:10:15,182 --> 00:10:18,118
"What parent wouldn't want
a true picture

253
00:10:18,185 --> 00:10:22,122
of the inner workings of
their child's mind and heart?"

254
00:10:22,189 --> 00:10:24,091
Well, that does sound appealing.

255
00:10:24,157 --> 00:10:25,492
On the other hand,

256
00:10:25,559 --> 00:10:27,294
until you realize
that the answer to this question

257
00:10:27,361 --> 00:10:29,997
is gonna be four letters long.

258
00:10:30,063 --> 00:10:35,302
Maybe that's a little bit
too much to ask of four letters.

259
00:10:37,871 --> 00:10:40,073
But Myers and Briggs
did not think it was.

260
00:10:40,140 --> 00:10:42,609
I mean, so I'll just say...

261
00:10:43,677 --> 00:10:46,713
We crave classification because
we don't like uncertainty.

262
00:10:46,780 --> 00:10:48,815
It makes us uncomfortable.
It makes us anxious.

263
00:10:48,882 --> 00:10:52,085
We wanna know
exactly what our child is like.

264
00:10:52,152 --> 00:10:54,054
We wanna know exactly
what they're gonna do.

265
00:10:54,121 --> 00:10:57,057
We wanna know exactly
how they're gonna react

266
00:10:57,124 --> 00:10:58,458
to any situation.

267
00:10:58,525 --> 00:11:00,661
And actually, I'm just gonna go
impromptu for a second here

268
00:11:00,727 --> 00:11:03,130
and say that in a classroom,
it's much the same, right?

269
00:11:03,197 --> 00:11:05,132
It would be great,
just as with our own children,

270
00:11:05,199 --> 00:11:08,335
it would be great if we could
divide our students into types

271
00:11:08,402 --> 00:11:13,006
and know exactly how each one
was gonna respond

272
00:11:13,073 --> 00:11:15,442
to everything we do
in the classroom

273
00:11:15,509 --> 00:11:19,746
via some very simple formula,
like four letters.

274
00:11:19,813 --> 00:11:21,014
It's probably not like that.

275
00:11:21,081 --> 00:11:22,616
But Myers and Briggs
certainly thought it was.

276
00:11:22,683 --> 00:11:25,319
They were very interesting
people.

277
00:11:25,385 --> 00:11:26,987
They got this classification
originally--

278
00:11:27,054 --> 00:11:29,189
They sort of modified it from
something that Carl Jung did,

279
00:11:29,256 --> 00:11:30,357
who they were obsessed with.

280
00:11:30,424 --> 00:11:32,593
Myers actually wrote

281
00:11:32,659 --> 00:11:35,662
an entire erotic fanfic novel
about Carl Jung,

282
00:11:35,729 --> 00:11:36,997
which I have not read.

283
00:11:37,064 --> 00:11:38,432
It's a little bit hard to find.

284
00:11:38,498 --> 00:11:39,833
Can I say erotic fanfic on PBS?

285
00:11:39,900 --> 00:11:41,502
I actually don't know
if that's allowed, but...

286
00:11:41,568 --> 00:11:43,537
[audience laughs]

287
00:11:44,571 --> 00:11:48,442
So, Briggs wrote,
very absolutistly,

288
00:11:48,509 --> 00:11:51,778
"Every one of us is born either
an extrovert or an introvert

289
00:11:51,845 --> 00:11:53,580
"and remains
extrovert and introvert

290
00:11:53,647 --> 00:11:54,781
to the end of his days."

291
00:11:54,848 --> 00:11:56,517
So, this very rigid
classification,

292
00:11:56,583 --> 00:11:58,485
just like Linnaeus.

293
00:11:58,552 --> 00:12:00,988
She also once described--
I mean, again,

294
00:12:01,054 --> 00:12:02,222
relating to this notion
of species,

295
00:12:02,289 --> 00:12:03,690
she also once described
a marriage

296
00:12:03,757 --> 00:12:05,759
of a T husband with an F wife

297
00:12:05,826 --> 00:12:08,328
as the marriage of a fish
and a canary.

298
00:12:08,395 --> 00:12:09,796
That's a pretty potent metaphor

299
00:12:09,863 --> 00:12:12,432
to just how incompatible
she thought these people were.

300
00:12:12,499 --> 00:12:15,502
So, maybe, okay,
I haven't said any math words.

301
00:12:15,569 --> 00:12:17,704
I guess I said 2 to the 4th.

302
00:12:17,771 --> 00:12:20,140
So, I think this kind of idea
already

303
00:12:20,207 --> 00:12:23,810
is, like, a great way of talking
about a word we like to say.

304
00:12:23,877 --> 00:12:28,382
I know we're doing a lot of
data science in this conference.

305
00:12:28,448 --> 00:12:30,017
So, already,
there's two different visions

306
00:12:30,083 --> 00:12:31,919
of what people are like
represented here.

307
00:12:31,985 --> 00:12:34,321
And we might call this
the bimodal model

308
00:12:34,388 --> 00:12:36,857
and the unimodal model, right?

309
00:12:36,924 --> 00:12:38,392
So, unimodal distributions,

310
00:12:38,458 --> 00:12:40,060
like,
have one big hump in the middle.

311
00:12:40,127 --> 00:12:42,296
That would be if most people
were kind of

312
00:12:42,362 --> 00:12:44,431
pretty roughly evenly divided

313
00:12:44,498 --> 00:12:45,766
between introverts
and extroverts,

314
00:12:45,832 --> 00:12:49,102
with a few extremes
on the outside.

315
00:12:49,169 --> 00:12:51,672
The bimodal model, which is what
Myers and Briggs believed in,

316
00:12:51,738 --> 00:12:54,842
was that there's really
two discrete types of people

317
00:12:54,908 --> 00:12:57,778
into which all people can
be divided, the Es and the Is,

318
00:12:57,845 --> 00:13:00,747
the extroverts and introverts.

319
00:13:00,814 --> 00:13:02,182
So, an interesting thing
happens.

320
00:13:02,249 --> 00:13:06,520
This test is devised and refined
in the 1930s and 1940s.

321
00:13:06,587 --> 00:13:08,989
In the '50s, when psychometrics
is at its peak,

322
00:13:09,056 --> 00:13:10,891
like, this really starts
to get some uptake.

323
00:13:10,958 --> 00:13:13,927
And in fact ETS,
a company we all work with,

324
00:13:13,994 --> 00:13:17,965
buys the MBTI and administers it
to millions of school children,

325
00:13:18,031 --> 00:13:19,132
'cause they're like, "Okay."

326
00:13:19,199 --> 00:13:20,567
They believed it.

327
00:13:20,634 --> 00:13:22,569
They're like, "We're gonna use
this to, like, improve education

328
00:13:22,636 --> 00:13:24,371
"and understand, like,
what kind of kids there are

329
00:13:24,438 --> 00:13:26,106
in the schools."

330
00:13:26,907 --> 00:13:29,243
And when they did that...

331
00:13:29,910 --> 00:13:31,912
they saw a lot of this.

332
00:13:32,913 --> 00:13:38,252
So, this is one of the many
charts in this report from ETS.

333
00:13:38,318 --> 00:13:39,920
I think this particular one

334
00:13:39,987 --> 00:13:42,222
shows liberal arts students
and engineering students,

335
00:13:42,289 --> 00:13:44,791
which in case you're interested,
their distribution

336
00:13:44,858 --> 00:13:46,460
of extrovert versus introvert
looks pretty similar.

337
00:13:46,527 --> 00:13:47,928
There's no big difference there.

338
00:13:47,995 --> 00:13:49,096
But the more important thing

339
00:13:49,162 --> 00:13:50,764
is this is a unimodal
distribution, right?

340
00:13:50,831 --> 00:13:52,699
There is no clear division.

341
00:13:52,766 --> 00:13:54,501
And actually, by the way,
this was the end

342
00:13:54,568 --> 00:13:56,303
of the association
of Myers and Briggs with ETS,

343
00:13:56,370 --> 00:13:58,105
because they were like,
"You've got to be wrong.

344
00:13:58,172 --> 00:13:59,439
"We know how the world is.

345
00:13:59,506 --> 00:14:01,608
"We know there's these two
separate types of people.

346
00:14:01,675 --> 00:14:04,511
We reject all of your statistics
and all of your analysis."

347
00:14:04,578 --> 00:14:08,682
They were like,
"This simply can't be right."

348
00:14:08,749 --> 00:14:10,918
I think nowadays
we do think this is right.

349
00:14:10,984 --> 00:14:12,719
I mean, there are lots
of psychological tests.

350
00:14:12,786 --> 00:14:14,254
And I think modern
personality psychology,

351
00:14:14,321 --> 00:14:16,089
which doesn't particularly use
the MBTI,

352
00:14:16,156 --> 00:14:18,692
uses other models
like the Big Five.

353
00:14:18,759 --> 00:14:19,893
In a different kind of talk,

354
00:14:19,960 --> 00:14:21,495
we could spend an hour, like,
talking about

355
00:14:21,562 --> 00:14:22,663
how dimensional analysis works

356
00:14:22,729 --> 00:14:24,097
and how modern personality
psychology

357
00:14:24,164 --> 00:14:26,834
chooses what dimensions
to measure people on.

358
00:14:26,900 --> 00:14:28,001
We're not gonna do that.

359
00:14:28,068 --> 00:14:29,169
I'm just gonna say

360
00:14:29,236 --> 00:14:33,240
that bimodality is out,
unimodality is in.

361
00:14:33,307 --> 00:14:36,109
But I'll just comment
that I want to say this

362
00:14:36,176 --> 00:14:38,078
without denying
the pull to classify,

363
00:14:38,145 --> 00:14:40,080
the pull to the bimodal vision,

364
00:14:40,147 --> 00:14:42,549
the pull to sort of separate
things into separate kinds.

365
00:14:42,616 --> 00:14:45,552
I mean, you even see it like--
You know, in political strategy,

366
00:14:45,619 --> 00:14:48,689
I just put up sort of two
popular political slogans,

367
00:14:48,755 --> 00:14:52,192
which I think sort of
are differently political-coded.

368
00:14:52,259 --> 00:14:53,594
One is, like,
a little more left-coded.

369
00:14:53,660 --> 00:14:54,828
One is a little more
right-coded.

370
00:14:54,895 --> 00:14:56,797
But it's interesting
to note that

371
00:14:56,864 --> 00:14:59,533
they both say the same thing.

372
00:15:00,467 --> 00:15:02,503
They both say
there are friends and enemies.

373
00:15:02,569 --> 00:15:03,904
There are two types of people,

374
00:15:03,971 --> 00:15:05,205
and they are distinct
from another.

375
00:15:05,272 --> 00:15:09,843
And if you are not one kind,
you are the other kind.

376
00:15:10,511 --> 00:15:12,746
I want to say one more thing
about the platypus.

377
00:15:12,813 --> 00:15:14,748
This is not gonna look like
it's about the platypus.

378
00:15:14,815 --> 00:15:17,885
But here's
a psychological question.

379
00:15:17,951 --> 00:15:19,586
Why are decisions hard?

380
00:15:19,653 --> 00:15:22,456
I think we've all experienced

381
00:15:22,823 --> 00:15:24,858
being stuck
with a difficult decision,

382
00:15:24,925 --> 00:15:28,862
two options that we can't decide
like, which one we like better.

383
00:15:28,929 --> 00:15:31,031
So, one model for this.

384
00:15:33,634 --> 00:15:35,936
Here's what Benjamin Franklin
said to do

385
00:15:36,003 --> 00:15:37,538
when you were faced
with a bad decision.

386
00:15:37,604 --> 00:15:40,140
He said, "Here's this thing
I like to do that I made up.

387
00:15:40,207 --> 00:15:42,709
"I like to, like,
take a piece of paper

388
00:15:42,776 --> 00:15:44,111
"and write a line
down the middle

389
00:15:44,178 --> 00:15:46,113
"and write all the reasons
in favor of one decision

390
00:15:46,180 --> 00:15:48,515
"on one side of the paper
and all the decisions,

391
00:15:48,582 --> 00:15:49,917
"all the reasons
for the other decision

392
00:15:49,983 --> 00:15:51,318
"on the other side of the paper,

393
00:15:51,385 --> 00:15:53,287
"and then sort of see
which column is bigger,

394
00:15:53,353 --> 00:15:55,455
like, which one
carries more weight."

395
00:15:55,522 --> 00:15:57,324
So, I am telling you

396
00:15:57,391 --> 00:15:59,159
that Benjamin Franklin
invented the pro-con list.

397
00:15:59,226 --> 00:16:01,061
I think this is actually true,

398
00:16:01,128 --> 00:16:03,263
'cause he writes about it
as something he came up with.

399
00:16:03,330 --> 00:16:04,665
And I can find
no earlier reference

400
00:16:04,731 --> 00:16:06,600
to anything of this kind.

401
00:16:06,667 --> 00:16:09,102
So, this is a model
where what's going on

402
00:16:09,169 --> 00:16:10,304
when you make a hard decision,

403
00:16:10,370 --> 00:16:14,875
it's hard because
you are balanced exactly 50/50.

404
00:16:14,942 --> 00:16:17,044
There's exactly 50% reasons
on one side

405
00:16:17,110 --> 00:16:19,046
and exactly 50% reasons
on the other side.

406
00:16:19,112 --> 00:16:22,683
And this kind of
continuous measurement--

407
00:16:22,749 --> 00:16:24,518
So, there's lots of
mathematical themes here

408
00:16:24,585 --> 00:16:26,386
that I'm just gonna skip past
very quickly.

409
00:16:26,453 --> 00:16:30,457
This is a question about
continuous versus discrete.

410
00:16:30,524 --> 00:16:31,992
And the continuous model,

411
00:16:32,059 --> 00:16:35,329
this kind of is how modern
personality psychology works.

412
00:16:35,395 --> 00:16:36,964
This is a picture
of somebody's measure--

413
00:16:37,030 --> 00:16:39,066
not mine, by the way.

414
00:16:39,132 --> 00:16:40,734
That's not a bad profile,
actually.

415
00:16:40,801 --> 00:16:42,936
Low on neuroticism--
Or low on agreeableness.

416
00:16:43,003 --> 00:16:45,906
Okay, it's not me.
[chuckles]

417
00:16:45,973 --> 00:16:48,375
But this, you know, nowadays
a personality psychology

418
00:16:48,442 --> 00:16:50,878
would say,
"Oh, if you can't decide

419
00:16:50,944 --> 00:16:52,112
"whether you're an extrovert
or introvert,

420
00:16:52,179 --> 00:16:53,547
"it's because
your score on the test

421
00:16:53,614 --> 00:16:55,015
"is you're really
very evenly poised--

422
00:16:55,082 --> 00:16:58,018
50% extrovert, 50% introvert."

423
00:16:58,085 --> 00:17:01,188
But here's the problem
with that model

424
00:17:01,255 --> 00:17:04,958
from the point of view
of making difficult decisions.

425
00:17:05,025 --> 00:17:08,829
If the issue were that you were
exactly balanced,

426
00:17:08,896 --> 00:17:11,231
exactly balanced
between the pros and the cons,

427
00:17:11,298 --> 00:17:14,434
exactly balanced
between the do this or do that,

428
00:17:14,501 --> 00:17:17,471
exactly balanced,
should I stay or should I go,

429
00:17:17,538 --> 00:17:20,007
then, if that were the case,

430
00:17:20,073 --> 00:17:21,742
then any new piece
of information

431
00:17:21,808 --> 00:17:23,844
that put more weight
on one side of the scale

432
00:17:23,911 --> 00:17:25,979
would suddenly make
the decision easy, right?

433
00:17:26,046 --> 00:17:28,182
It would tip the scales.

434
00:17:28,248 --> 00:17:30,017
And what I think--
And, actually,

435
00:17:30,083 --> 00:17:31,652
'cause I'm sort of working
on this material,

436
00:17:31,718 --> 00:17:32,853
I'm actually curious
if people agree.

437
00:17:32,920 --> 00:17:34,454
We can talk in the Q and A.

438
00:17:34,521 --> 00:17:36,657
I think that hard decisions
that we face in real life

439
00:17:36,723 --> 00:17:38,458
typically do not feel like that.

440
00:17:38,525 --> 00:17:40,961
I think when we're really stuck

441
00:17:41,028 --> 00:17:43,664
and you get a new piece
of information or a new fact

442
00:17:43,730 --> 00:17:45,265
that weighs to one side
or the other,

443
00:17:45,332 --> 00:17:47,267
we're often still kind of stuck.

444
00:17:47,334 --> 00:17:48,635
And I think these decisions,

445
00:17:48,702 --> 00:17:52,873
they're a little bit more
like the platypus.

446
00:17:52,940 --> 00:17:54,408
Like, it is hard to figure out

447
00:17:54,474 --> 00:17:56,543
whether we should think
the platypus is a mammal

448
00:17:56,610 --> 00:17:59,079
or whether we should think
platypus is a bird.

449
00:17:59,146 --> 00:18:03,717
But if the platypus
suddenly became 10% hairier,

450
00:18:03,784 --> 00:18:06,787
that would not settle
the matter, right?

451
00:18:06,854 --> 00:18:08,755
So, I think the issue here
is somehow not

452
00:18:08,822 --> 00:18:10,724
that the platypus
is exactly balanced,

453
00:18:10,791 --> 00:18:12,793
50% mammal, 50% bird.

454
00:18:14,194 --> 00:18:16,730
I think it's that the platypus
has certain features

455
00:18:16,797 --> 00:18:19,600
which we identify as definitely
mammal and certain features,

456
00:18:19,666 --> 00:18:21,869
like a bill, which we identify
as definitely bird,

457
00:18:21,935 --> 00:18:25,372
and a slight perturbation
simply does not change that.

458
00:18:25,439 --> 00:18:27,241
It leaves us
in the same position.

459
00:18:27,307 --> 00:18:31,211
So, I think that these rough
psychological decisions,

460
00:18:31,278 --> 00:18:32,446
problems maybe,
are more like that,

461
00:18:32,513 --> 00:18:33,847
that there are sort of
some features

462
00:18:33,914 --> 00:18:35,249
that we associate
with good decisions

463
00:18:35,315 --> 00:18:37,251
and some features
that we associate bad decisions,

464
00:18:37,317 --> 00:18:39,987
and the things that we truly
get stuck on are the things

465
00:18:40,053 --> 00:18:41,555
that have features of each,

466
00:18:41,622 --> 00:18:43,857
things where we simply
do not solve the problem

467
00:18:43,924 --> 00:18:48,495
by making the problem
10% hairier or 10% smoother.

468
00:18:49,930 --> 00:18:53,200
And at this point,
20 minutes into the talk,

469
00:18:53,267 --> 00:18:54,868
I've talked about
all these difficult things,

470
00:18:54,935 --> 00:18:56,737
things we're really stuck on.

471
00:18:56,803 --> 00:18:59,339
And maybe we find ourselves
saying, as math teachers,

472
00:18:59,406 --> 00:19:03,377
"If only everything
could be like math."

473
00:19:03,443 --> 00:19:05,078
I mean math, right,
we all do it,

474
00:19:05,145 --> 00:19:07,915
math is the land
where we actually do have

475
00:19:07,981 --> 00:19:11,018
these extremely nice,
clean classifications

476
00:19:11,084 --> 00:19:13,854
with definitions
that are unequivocal

477
00:19:13,921 --> 00:19:16,190
and make sense
and tell us what is what.

478
00:19:16,256 --> 00:19:17,391
I don't know about you guys,

479
00:19:17,457 --> 00:19:18,892
but, like,
I do actually find mathematics

480
00:19:18,959 --> 00:19:21,161
like, rather psychologically
soothing,

481
00:19:21,228 --> 00:19:22,763
like a sort of nice place to be

482
00:19:22,829 --> 00:19:24,798
when, like, other stuff
is confusing.

483
00:19:24,865 --> 00:19:26,433
I think probably
we've all had those feelings.

484
00:19:26,500 --> 00:19:28,335
And in some sense,

485
00:19:28,402 --> 00:19:32,906
the fundamental classification
of numbers into odd and even,

486
00:19:32,973 --> 00:19:35,175
something that many of you
have probably taught, right?

487
00:19:35,242 --> 00:19:38,612
Do I have some elementary school
teachers in the audience today?

488
00:19:38,679 --> 00:19:41,215
So, this is, like, one of the
most basic classifications

489
00:19:41,281 --> 00:19:42,382
in mathematics.

490
00:19:42,449 --> 00:19:43,550
And I'm a number theorist,

491
00:19:43,617 --> 00:19:45,152
so I think about this
all the time.

492
00:19:45,219 --> 00:19:47,421
Here's a definition
that you might have in mind.

493
00:19:47,487 --> 00:19:48,689
A number is even

494
00:19:48,755 --> 00:19:50,691
if you can divide the things
into two equal parts,

495
00:19:50,757 --> 00:19:51,959
like shoes, right?

496
00:19:52,025 --> 00:19:54,161
The left shoes and the
right shoes are equal in number.

497
00:19:54,228 --> 00:19:55,362
And odd if you can't,

498
00:19:55,429 --> 00:19:59,266
if you try to do that
and there's one left over.

499
00:19:59,333 --> 00:20:02,236
But already there's a problem.

500
00:20:04,037 --> 00:20:06,840
The problem has to do with
our favorite number, zero.

501
00:20:06,907 --> 00:20:10,143
And what you are looking at
is a picture of a gas line

502
00:20:10,210 --> 00:20:14,448
during a gas-rationing,
energy-cost spike in the '80s.

503
00:20:14,515 --> 00:20:16,650
Why am I showing you
a picture of a gas line?

504
00:20:16,717 --> 00:20:19,219
Because when there's
a gas shortage,

505
00:20:19,286 --> 00:20:21,722
as you may know
if you've lived through one,

506
00:20:21,788 --> 00:20:25,259
a city or state or local
government will usually say,

507
00:20:25,325 --> 00:20:27,694
"Okay, we can't have everybody
going to the gas station at once

508
00:20:27,761 --> 00:20:29,296
"'cause the lines will be
too long.

509
00:20:29,363 --> 00:20:30,998
"So, everybody whose license
number,

510
00:20:31,064 --> 00:20:32,399
"plate number
ends with an even number,

511
00:20:32,466 --> 00:20:34,201
"you can go
on these days of the week.

512
00:20:34,268 --> 00:20:38,038
"And if your license plate
number ends with an odd number,

513
00:20:38,105 --> 00:20:39,973
you go on the other days
of the week."

514
00:20:40,040 --> 00:20:41,875
And every single time
this happens--

515
00:20:41,942 --> 00:20:44,211
and I've actually read
lots of examples of this,

516
00:20:44,278 --> 00:20:46,346
people who have a license plate
that ends in zero

517
00:20:46,413 --> 00:20:50,417
totally freak out
and do not know what to do.

518
00:20:51,285 --> 00:20:52,653
And there's a huge conflict
about this,

519
00:20:52,719 --> 00:20:55,589
so much so that in fact that
in my home state of Maryland,

520
00:20:55,656 --> 00:20:57,291
where I grew up,

521
00:20:57,357 --> 00:21:01,695
zero is an even number
by state statute. [all laugh]

522
00:21:01,762 --> 00:21:03,530
So, they had to put this
in the law

523
00:21:03,597 --> 00:21:05,365
because there were so many
arguments in the '70s

524
00:21:05,432 --> 00:21:07,367
about, like, when you could go
to the gas station.

525
00:21:07,434 --> 00:21:09,603
And now, once you cross
the border into Maryland,

526
00:21:09,670 --> 00:21:11,138
zero becomes an even number,

527
00:21:11,205 --> 00:21:14,308
and it's illegal for zero
not to be an even number.

528
00:21:14,374 --> 00:21:16,043
So...

529
00:21:16,476 --> 00:21:18,478
Why do I bring this up?

530
00:21:19,413 --> 00:21:20,514
One tone of voice

531
00:21:20,581 --> 00:21:22,716
in which I could be
bringing this up is to say,

532
00:21:22,783 --> 00:21:24,484
"Boy, isn't it sad
that people don't know

533
00:21:24,551 --> 00:21:26,286
"what an odd number
or an even number is?

534
00:21:26,353 --> 00:21:27,888
"Like, what are we doing
as math educators

535
00:21:27,955 --> 00:21:29,156
that people don't know this?"

536
00:21:29,223 --> 00:21:33,160
That is not the tone of voice
I wish to adopt,

537
00:21:33,227 --> 00:21:34,461
because I think--

538
00:21:34,528 --> 00:21:36,697
and this is one of my deepest
beliefs as a teacher--

539
00:21:36,763 --> 00:21:39,366
that when people
are confused about a thing,

540
00:21:39,433 --> 00:21:42,569
it is because the thing
is confusing.

541
00:21:42,636 --> 00:21:44,404
Not because we've failed,

542
00:21:44,471 --> 00:21:48,475
but because there is
an actual difficulty.

543
00:21:48,742 --> 00:21:53,247
And here, I think the difficulty
is in the definition, because--

544
00:21:53,313 --> 00:21:55,315
And now,
because this has happened in,

545
00:21:55,382 --> 00:21:58,018
and we haven't had lines like
that in New York or Maryland

546
00:21:58,085 --> 00:21:59,186
in quite a few years.

547
00:21:59,253 --> 00:22:01,288
But it does happen
around the world,

548
00:22:01,355 --> 00:22:02,689
and this is
a cross-cultural phenomenon,

549
00:22:02,756 --> 00:22:04,958
and you can, like,
read people's tweets about it

550
00:22:05,025 --> 00:22:06,226
when they're upset
about this thing.

551
00:22:06,293 --> 00:22:08,829
And you can see people saying,

552
00:22:08,896 --> 00:22:12,733
"Why are they saying
zero is an even number?"

553
00:22:12,799 --> 00:22:16,136
You can't divide nothing
into two equal parts.

554
00:22:16,203 --> 00:22:17,337
Like, how can you do it?

555
00:22:17,404 --> 00:22:19,506
There's nothing to divide.

556
00:22:19,573 --> 00:22:23,510
That, guys, is a good objection,
I believe.

557
00:22:23,577 --> 00:22:26,547
If your definition
is that an even number

558
00:22:26,613 --> 00:22:28,882
is when that number of things

559
00:22:28,949 --> 00:22:30,284
can be divided
into two equal parts,

560
00:22:30,350 --> 00:22:31,919
I think it is perfectly
reasonable to say,

561
00:22:31,985 --> 00:22:33,120
"That is something I cannot do

562
00:22:33,187 --> 00:22:35,489
if there's nothing there
to divide."

563
00:22:35,556 --> 00:22:36,990
So, zero cannot be even.

564
00:22:37,057 --> 00:22:39,026
Now, often,
people in that situation--

565
00:22:39,092 --> 00:22:41,428
And actually, there are, like,
wonderful studies of this

566
00:22:41,495 --> 00:22:43,430
where they interview children
and sort of talk to them

567
00:22:43,497 --> 00:22:44,831
about their notions
of odd and even.

568
00:22:44,898 --> 00:22:47,034
Often people will say that
zero is neither odd nor even,

569
00:22:47,100 --> 00:22:48,869
because you can't divide it
into two equal parts,

570
00:22:48,936 --> 00:22:50,137
but it's also not the case

571
00:22:50,204 --> 00:22:52,339
that if you try to divide it
into two equal parts,

572
00:22:52,406 --> 00:22:53,507
there's one left over.

573
00:22:53,574 --> 00:22:55,642
There is simply nothing to try.

574
00:22:55,709 --> 00:22:58,579
And so, people would see zero--

575
00:22:58,645 --> 00:23:00,981
many children and adults, like,
sort of see zero as a number

576
00:23:01,048 --> 00:23:03,717
that is neither odd nor even.

577
00:23:03,784 --> 00:23:07,254
So, you know, we would say--
[laughs]

578
00:23:08,121 --> 00:23:10,324
As grown-up mathematicians,
we would say, "Okay,

579
00:23:10,390 --> 00:23:11,491
"but the problem is

580
00:23:11,558 --> 00:23:13,827
"that's not the definition
of an even number.

581
00:23:13,894 --> 00:23:15,863
"It's not this folk definition

582
00:23:15,929 --> 00:23:18,532
"that the number of things can
be divided into two equal parts.

583
00:23:18,599 --> 00:23:23,303
"It's a more algebraic
definition that a number is even

584
00:23:23,370 --> 00:23:27,040
when it is twice
another whole number."

585
00:23:27,574 --> 00:23:29,142
And in that case, like,
you're fine, right?

586
00:23:29,209 --> 00:23:30,878
You can say,
"Well, zero is even

587
00:23:30,944 --> 00:23:33,013
"simply because
it is two times zero

588
00:23:33,080 --> 00:23:34,214
and zero
is a whole number."

589
00:23:34,281 --> 00:23:35,582
It even applies
to negative numbers,

590
00:23:35,649 --> 00:23:37,417
where then-- I mean,
nobody's license plate

591
00:23:37,484 --> 00:23:38,819
ends with a negative number,
thank God.

592
00:23:38,886 --> 00:23:41,555
But if it did, people would be
even more confused, right?

593
00:23:41,622 --> 00:23:45,125
'Cause I think the second
definition applies very well

594
00:23:45,192 --> 00:23:46,326
to negative numbers,

595
00:23:46,393 --> 00:23:47,928
and to the first one,
it does not.

596
00:23:47,995 --> 00:23:50,430
So hold this
in your mind.

597
00:23:50,497 --> 00:23:52,232
I'm gonna say
a little bit more about this,

598
00:23:52,299 --> 00:23:54,434
but I wanna sort of go
to some other places first

599
00:23:54,501 --> 00:23:55,969
and then come back to it.

600
00:23:56,036 --> 00:23:57,171
I just hope to convince you

601
00:23:57,237 --> 00:23:58,472
that there's
an actual difficulty here,

602
00:23:58,539 --> 00:23:59,673
so there's something
to think about

603
00:23:59,740 --> 00:24:01,408
when we think about
what is the definition

604
00:24:01,475 --> 00:24:03,944
of an odd
or an even number.

605
00:24:04,444 --> 00:24:06,980
Maybe this will be, like,
a slight discursive,

606
00:24:07,047 --> 00:24:08,382
but just for a minute.

607
00:24:08,448 --> 00:24:13,654
I mean, we are kind of in this
great age of classification

608
00:24:13,720 --> 00:24:16,390
where, you know, already...

609
00:24:17,157 --> 00:24:18,659
I would say 20 years ago,

610
00:24:18,725 --> 00:24:20,928
we could already train a neural
network, like, very well

611
00:24:20,994 --> 00:24:24,331
to classify images
into dog and cat.

612
00:24:27,668 --> 00:24:32,072
And it does this with a great
deal of nonambiguity.

613
00:24:32,139 --> 00:24:34,541
It does not--
There are not a lot of images

614
00:24:34,608 --> 00:24:38,645
to which it would say
it's 50% dog, 50% cat.

615
00:24:38,712 --> 00:24:40,981
It pretty much--
a well-trained neural net

616
00:24:41,048 --> 00:24:42,616
will always choose one
or the other.

617
00:24:42,683 --> 00:24:44,218
This is, like,
a little bit tangential

618
00:24:44,284 --> 00:24:46,820
to the main point of this talk,
but I wanna bring it up,

619
00:24:46,887 --> 00:24:49,223
because I think
not everybody knows it.

620
00:24:49,289 --> 00:24:51,358
The way a large language model
is trained today,

621
00:24:51,425 --> 00:24:52,893
any of the commercial
large language models

622
00:24:52,960 --> 00:24:55,963
you would interact with
through a chat,

623
00:24:56,029 --> 00:25:00,567
they are in fact trained
to give very definitive answers.

624
00:25:00,634 --> 00:25:02,135
They are trained to be sure.

625
00:25:02,202 --> 00:25:05,005
They are trained to sort of...

626
00:25:05,405 --> 00:25:07,975
not be very ambiguous
about what a thing is.

627
00:25:08,041 --> 00:25:09,576
And the reason
they trained it that way

628
00:25:09,643 --> 00:25:13,213
is because they do want them
to give correct factual answers.

629
00:25:13,280 --> 00:25:15,182
So, in some sense...

630
00:25:16,216 --> 00:25:17,818
You know, if you ask, like,

631
00:25:17,885 --> 00:25:19,152
"What's the capital
of Wisconsin?"

632
00:25:19,219 --> 00:25:20,754
You actually don't want
the large language model

633
00:25:20,821 --> 00:25:24,591
to do something like imitate
what a random person would say

634
00:25:24,658 --> 00:25:26,827
when asked, "What's the capital
of Wisconsin?"

635
00:25:26,894 --> 00:25:29,162
Because I can tell you, being
originally from the East Coast,

636
00:25:29,229 --> 00:25:30,564
like, lots of people in Maryland

637
00:25:30,631 --> 00:25:32,699
do not know what the capital
of Wisconsin is

638
00:25:32,766 --> 00:25:36,937
or, like, where Wisconsin is
or anything about Wisconsin.

639
00:25:37,004 --> 00:25:38,539
I think the people
who build these models,

640
00:25:38,605 --> 00:25:40,474
like, want them to reliably give
the right answer,

641
00:25:40,541 --> 00:25:42,476
which means that they want
to sort of train them

642
00:25:42,543 --> 00:25:44,978
to be very, very definitive

643
00:25:45,045 --> 00:25:49,116
about what the exact answer
to each question is.

644
00:25:49,183 --> 00:25:50,851
And that creates,
like, some weird behavior.

645
00:25:50,918 --> 00:25:52,653
This is sort of one
of my favorite runs,

646
00:25:52,719 --> 00:25:55,222
where if you sort of--
this is GPT-3.5.

647
00:25:55,289 --> 00:25:58,292
If you ask, "Give an example
of a Jewish person--"

648
00:25:58,358 --> 00:25:59,960
And this is true
for almost every model.

649
00:26:00,027 --> 00:26:01,929
You will almost always
get Einstein.

650
00:26:01,995 --> 00:26:03,263
Like, according to modern AI,

651
00:26:03,330 --> 00:26:06,834
Albert Einstein
was the only Jew in history.

652
00:26:06,900 --> 00:26:09,436
[audience laughs]
He comes up almost every time.

653
00:26:09,503 --> 00:26:11,705
Albert Einstein is about--
depends on the model.

654
00:26:11,772 --> 00:26:13,574
But about 90% to 95%

655
00:26:13,907 --> 00:26:15,509
of the world
historical Jewish population

656
00:26:15,576 --> 00:26:16,677
is Albert Einstein.

657
00:26:16,743 --> 00:26:17,845
In this particular one,

658
00:26:17,911 --> 00:26:20,280
I got 19 Einsteins
and 1 Ruth Bader Ginsburg.

659
00:26:20,347 --> 00:26:23,817
And that's pretty typical,
you know?

660
00:26:25,519 --> 00:26:26,620
Okay, but that's an aside.

661
00:26:26,687 --> 00:26:30,023
I wanna come back to math,
actually.

662
00:26:31,725 --> 00:26:34,528
So, we're gonna go
from arithmetic to calculus.

663
00:26:34,595 --> 00:26:38,232
What you're looking at here
is a page from Fermat

664
00:26:38,298 --> 00:26:41,401
where he's writing about
how to compute the area

665
00:26:41,468 --> 00:26:43,470
inside a curved region.

666
00:26:44,838 --> 00:26:46,773
And he says, "I compare this
to the first solid,

667
00:26:46,840 --> 00:26:48,141
"a squared b - a cubed,

668
00:26:48,208 --> 00:26:52,212
as if they were equal,
although in fact they are not."

669
00:26:52,279 --> 00:26:53,947
And I put this up

670
00:26:54,014 --> 00:26:57,117
because I think
this is the kind of sentence

671
00:26:57,184 --> 00:26:59,786
that most people
outside mathematics

672
00:26:59,853 --> 00:27:02,789
think that mathematicians
would not write.

673
00:27:02,856 --> 00:27:04,992
In fact, I think this is
the exact kind of sentence

674
00:27:05,058 --> 00:27:07,594
they think that our entire job
is not to write,

675
00:27:07,661 --> 00:27:11,331
not to say, "Well, let's act
as if these things are equal,

676
00:27:11,398 --> 00:27:12,499
even though they're not."

677
00:27:12,566 --> 00:27:14,735
You know, a normal person
outside of our world

678
00:27:14,801 --> 00:27:17,938
would say, like, "I thought
your guys' entire thing

679
00:27:18,005 --> 00:27:19,806
"was to say things were equal
when they're equal

680
00:27:19,873 --> 00:27:23,143
and say they're not equal
when they're not equal."

681
00:27:23,210 --> 00:27:26,280
And yet, here is Fermat,
a great mathematician,

682
00:27:26,346 --> 00:27:27,581
saying just the opposite.

683
00:27:27,648 --> 00:27:30,951
"We're gonna ask--
We're gonna act as if

684
00:27:31,018 --> 00:27:32,786
these numbers are equal
when they're not,"

685
00:27:32,853 --> 00:27:35,889
because they differ
in some sense infinitesimally.

686
00:27:35,956 --> 00:27:40,260
And of course, this is what the
entire history--

687
00:27:40,327 --> 00:27:43,030
at least the early history
of calculus is like.

688
00:27:43,096 --> 00:27:44,331
Calculus runs on vibes.

689
00:27:44,398 --> 00:27:46,500
This is a fact, right?

690
00:27:46,567 --> 00:27:49,703
Certainly,
Isaac Newton was no more--

691
00:27:49,770 --> 00:27:53,040
He said stuff like this
constantly,

692
00:27:53,106 --> 00:27:54,241
to the point that, you know,

693
00:27:54,308 --> 00:27:55,642
he was criticized
by people around him.

694
00:27:55,709 --> 00:27:57,077
Famously, Bishop Berkeley

695
00:27:57,144 --> 00:27:59,880
wrote a very angry
anti-Newton screed saying,

696
00:27:59,947 --> 00:28:01,748
"What are these differences,
Newton,

697
00:28:01,815 --> 00:28:05,185
"that you write about that like,
'Well, they're not zero.

698
00:28:05,252 --> 00:28:06,353
"They're infinitely small.

699
00:28:06,420 --> 00:28:07,521
"They're smaller than any
number.

700
00:28:07,588 --> 00:28:09,122
But at the same time,
they're not zero."

701
00:28:09,189 --> 00:28:10,324
And he's like, "What are they,

702
00:28:10,390 --> 00:28:11,491
the ghosts
of departed quantities?"

703
00:28:11,558 --> 00:28:14,661
[audience laughs]
Famous, famous Newton insult.

704
00:28:14,728 --> 00:28:17,264
And the funny thing about it is
that Berkeley was kind of right.

705
00:28:17,331 --> 00:28:18,465
He kind of had a point.

706
00:28:18,532 --> 00:28:21,468
We can't deny it
from a modern point of view.

707
00:28:21,535 --> 00:28:22,736
And you could ask, well,

708
00:28:22,803 --> 00:28:26,607
"So, did Isaac Newton
actually know calculus?"

709
00:28:26,673 --> 00:28:28,342
He could not have written
an epsilon-delta proof

710
00:28:28,408 --> 00:28:31,712
like we write it today,
those of us who teach calculus.

711
00:28:31,778 --> 00:28:33,580
So, here, I know this talk
is about uncertainty,

712
00:28:33,647 --> 00:28:36,149
but I'm gonna give a definitive
answer to this question.

713
00:28:36,216 --> 00:28:38,051
Yes, Isaac Newton
invented calculus.

714
00:28:38,118 --> 00:28:39,953
He definitely knew calculus.

715
00:28:40,020 --> 00:28:41,889
I feel like it's absurd to say

716
00:28:41,955 --> 00:28:43,490
that Isaac Newton
didn't know calculus,

717
00:28:43,557 --> 00:28:45,425
Newton and Leibniz.

718
00:28:45,492 --> 00:28:47,761
He didn't know fully
how to formalize it,

719
00:28:47,828 --> 00:28:48,962
but he knew what it was.

720
00:28:49,029 --> 00:28:50,364
He knew what he was doing.

721
00:28:50,430 --> 00:28:52,299
I feel like I refuse to accept

722
00:28:52,366 --> 00:28:56,170
an alternative answer
to this question.

723
00:28:57,704 --> 00:29:01,108
It's not until
more than 100 years later,

724
00:29:01,175 --> 00:29:02,709
in the early 19th century,

725
00:29:02,776 --> 00:29:05,112
that Cauchy invents, like,
the notion of a limit

726
00:29:05,179 --> 00:29:07,114
that we teach in AP calculus
today,

727
00:29:07,181 --> 00:29:09,550
the notion of the epsilon-delta
definition of continuity.

728
00:29:09,616 --> 00:29:11,518
All this stuff
is, like, brand-new.

729
00:29:11,585 --> 00:29:14,688
And by the way, Cauchy gets
very excited about it

730
00:29:14,755 --> 00:29:18,192
and immediately changes what
he's teaching the freshman

731
00:29:18,258 --> 00:29:21,161
at the �cole normale in Paris,
he's teaching the engineers.

732
00:29:21,228 --> 00:29:22,329
And he immediately is like,

733
00:29:22,396 --> 00:29:24,298
"I finally figured out
what calculus really is."

734
00:29:24,364 --> 00:29:26,567
And, then, it created,
like, a huge revolt,

735
00:29:26,633 --> 00:29:28,969
and in the end, like, the dean
had to be brought in

736
00:29:29,036 --> 00:29:30,704
to sort of sit
in Cauchy's classroom

737
00:29:30,771 --> 00:29:33,440
to make sure he was
not teaching epsilon-delta

738
00:29:33,507 --> 00:29:35,576
to the freshman engineers

739
00:29:35,642 --> 00:29:37,244
who were, like,
practically rioting

740
00:29:37,311 --> 00:29:39,513
at this sort of bizarre novelty
that Cauchy had invented.

741
00:29:39,580 --> 00:29:43,350
So, I'll just say I like
this quote from Emanuel Lasker,

742
00:29:43,417 --> 00:29:46,053
who was not a mathematician,
but a chess player.

743
00:29:46,119 --> 00:29:48,355
But I think this is very
appropriate to Newton.

744
00:29:48,422 --> 00:29:50,591
He says, "He who relies
solely upon tactics

745
00:29:50,657 --> 00:29:52,092
"that he can
wholly comprehend

746
00:29:52,159 --> 00:29:53,927
"is liable,
in the course of time,

747
00:29:53,994 --> 00:29:56,530
to weaken his imagination."

748
00:29:56,597 --> 00:29:57,764
So, very true.

749
00:29:57,831 --> 00:29:59,566
We can say that there were
things about calculus

750
00:29:59,633 --> 00:30:00,734
that Newton didn't understand.

751
00:30:00,801 --> 00:30:02,336
But to say that Newton
didn't know calculus,

752
00:30:02,402 --> 00:30:03,704
even though he was saying

753
00:30:03,770 --> 00:30:05,506
imprecise things like, "Let's
treat this number as zero,

754
00:30:05,572 --> 00:30:06,673
even though it's not,"

755
00:30:06,740 --> 00:30:07,841
breaking
this kind of classification

756
00:30:07,908 --> 00:30:09,009
that we think

757
00:30:09,076 --> 00:30:10,944
is so characteristic
of mathematicians,

758
00:30:11,011 --> 00:30:13,680
I think to say he did not know
calculus is absurd.

759
00:30:13,747 --> 00:30:15,749
Actually, one of my favorite
quotes on this

760
00:30:15,816 --> 00:30:20,387
is from the topologist Poincar�,
who again says something

761
00:30:20,454 --> 00:30:21,722
I don't think people outside

762
00:30:21,788 --> 00:30:23,223
think of mathematicians
as saying,

763
00:30:23,290 --> 00:30:24,825
that "Mathematics is the art

764
00:30:24,892 --> 00:30:28,929
of giving the same name
to different things."

765
00:30:28,996 --> 00:30:32,766
I mean, this would make
Linnaeus puke, right?

766
00:30:32,833 --> 00:30:37,070
Like, this is the exact opposite
point of view.

767
00:30:37,137 --> 00:30:39,540
But I think Poincar� had this
as a hard-won insight,

768
00:30:39,606 --> 00:30:41,375
that it is not always the case

769
00:30:41,441 --> 00:30:42,910
that the actual things
in the world--

770
00:30:42,976 --> 00:30:46,313
and for me, the mathematical
world is part of the world--

771
00:30:46,380 --> 00:30:48,649
that they cleave naturally
into types.

772
00:30:48,715 --> 00:30:50,817
And I think partly,
Poincar� was writing this

773
00:30:50,884 --> 00:30:52,252
under the influence
of the development

774
00:30:52,319 --> 00:30:54,154
of non-Euclidean geometry,

775
00:30:54,221 --> 00:30:56,156
another story we could spend
a whole hour talking about,

776
00:30:56,223 --> 00:30:57,691
which I'm not gonna.

777
00:30:57,758 --> 00:31:00,427
But at the time
Poincar� was writing this,

778
00:31:00,494 --> 00:31:02,162
standard lines
in Euclid's plane,

779
00:31:02,229 --> 00:31:03,497
a great circle on a sphere--

780
00:31:03,564 --> 00:31:05,399
And that's my incredibly bad
picture

781
00:31:05,465 --> 00:31:08,869
of geodesic rays
on a hyperbolic disk.

782
00:31:08,936 --> 00:31:12,005
All of these were lines
in various versions of geometry,

783
00:31:12,072 --> 00:31:14,308
and all deserved
to be called lines,

784
00:31:14,374 --> 00:31:16,944
as Poincar� knew very well.

785
00:31:17,411 --> 00:31:19,246
Let's do a little more calculus.

786
00:31:19,313 --> 00:31:21,949
And I'm gonna do this
a little bit speedy.

787
00:31:22,015 --> 00:31:24,251
Here is something
that, at one time,

788
00:31:24,318 --> 00:31:26,453
was a very serious
mathematical controversy.

789
00:31:26,520 --> 00:31:28,956
What is the sum
of this infinite series,

790
00:31:29,022 --> 00:31:32,593
1 - 1 + 1 - 1 + 1 - 1,
dot, dot, dot, forever.

791
00:31:35,562 --> 00:31:36,864
There's an obvious answer,

792
00:31:36,930 --> 00:31:38,665
which is we can kind of
PEMDAS this out

793
00:31:38,732 --> 00:31:41,401
and put in some parentheses
and write it like this.

794
00:31:41,468 --> 00:31:44,471
And it seems to be, like, the
sum of infinitely many zeros.

795
00:31:44,538 --> 00:31:47,508
So, okay, obviously it's zero.

796
00:31:48,876 --> 00:31:51,378
On the other hand,
we can group a different way

797
00:31:51,445 --> 00:31:55,315
to make it 1 - 0 - 0 - 0 - 0,
et cetera, et cetera,

798
00:31:55,382 --> 00:31:57,117
from which it seems to be
obviously 1.

799
00:31:57,184 --> 00:31:58,952
This is called
the Grandi's series, by the way,

800
00:31:59,019 --> 00:32:01,855
'cause Grandi was a sort of
somewhat unhinged monk

801
00:32:01,922 --> 00:32:04,458
in the 17th century who, like,
wrote a lot about this.

802
00:32:04,525 --> 00:32:07,060
And Grandi's answer, by the way,
if we had a little more time

803
00:32:07,127 --> 00:32:08,462
we would, like,
really algebra this out,

804
00:32:08,529 --> 00:32:12,332
but I'm just gonna show you
one slide for the algebra fans.

805
00:32:12,399 --> 00:32:15,269
Grandi argued that you could
show rather easily

806
00:32:15,335 --> 00:32:18,272
that if X is the sum,
then X is equal to 1 - X.

807
00:32:18,338 --> 00:32:20,007
So, X had to be 1/2.

808
00:32:21,375 --> 00:32:23,443
Which is maybe
the weirdest answer of all,

809
00:32:23,510 --> 00:32:27,714
like, that an infinite sum of
whole numbers would be 1/2.

810
00:32:27,781 --> 00:32:29,716
But Grandi was very committed
to this point of view.

811
00:32:29,783 --> 00:32:31,185
And people really argued
about this.

812
00:32:31,251 --> 00:32:33,120
And I just want to...

813
00:32:37,357 --> 00:32:39,793
I just wanna give this quote
from Hardy, G.H. Hardy,

814
00:32:39,860 --> 00:32:40,961
the great number theorist

815
00:32:41,028 --> 00:32:43,530
of the early 20th century
in England.

816
00:32:43,597 --> 00:32:46,200
I am gonna actually read this
out, 'cause it's important.

817
00:32:46,266 --> 00:32:48,468
"It does not occur--"
By the way,

818
00:32:48,535 --> 00:32:50,270
think about odd and even
as I read this.

819
00:32:50,337 --> 00:32:52,072
This is bringing us back
to odd and even.

820
00:32:52,139 --> 00:32:53,674
"It does not occur
to a modern mathematician

821
00:32:53,740 --> 00:32:55,142
"that a collection
of mathematical symbols

822
00:32:55,209 --> 00:32:56,443
"should have a 'meaning'

823
00:32:56,510 --> 00:32:59,880
"until one has been
assigned to it by definition.

824
00:32:59,947 --> 00:33:01,048
"It was not a triviality

825
00:33:01,114 --> 00:33:02,382
"even to
the greatest mathematicians

826
00:33:02,449 --> 00:33:03,617
"of the 18th Century.

827
00:33:03,684 --> 00:33:07,287
"They had not the habit
of definition.

828
00:33:07,354 --> 00:33:08,488
"It was not natural to them

829
00:33:08,555 --> 00:33:11,425
"to say in so many words,
'By X we mean Y.'

830
00:33:11,491 --> 00:33:12,626
"It is broadly true to say

831
00:33:12,693 --> 00:33:14,394
that mathematicians
before Cauchy--"

832
00:33:14,461 --> 00:33:16,129
The bad calculus teacher,
remember him?

833
00:33:16,196 --> 00:33:17,998
"-- mathematicians
before Cauchy asked not,

834
00:33:18,065 --> 00:33:19,299
"'How shall we define

835
00:33:19,366 --> 00:33:20,567
"the sum
of this infinite series?',

836
00:33:20,634 --> 00:33:23,937
"but, 'What is the sum
of this infinite series?'

837
00:33:24,004 --> 00:33:25,806
"And that this
habit of mind led them

838
00:33:25,873 --> 00:33:28,709
"into unnecessary
perplexities and controversies

839
00:33:28,775 --> 00:33:31,912
which were often really verbal."

840
00:33:34,248 --> 00:33:38,185
This, again, is counter
to most people's view

841
00:33:38,252 --> 00:33:40,621
of what
a mathematician's job is.

842
00:33:40,687 --> 00:33:45,292
I think they do think it is
our job to say what things are.

843
00:33:45,359 --> 00:33:46,760
And I think
the modern mathematical view,

844
00:33:46,827 --> 00:33:51,198
as evidenced by Hardy here,
is that's just not right.

845
00:33:51,265 --> 00:33:52,599
We're here to sort of figure out

846
00:33:52,666 --> 00:33:56,003
what things
should be defined to be.

847
00:33:57,504 --> 00:33:58,705
It's a big philosophical shift.

848
00:33:58,772 --> 00:34:00,974
I wanna say
a little bit more about it.

849
00:34:01,041 --> 00:34:02,142
I'll just comment.

850
00:34:02,209 --> 00:34:03,310
I don't wanna say, like,

851
00:34:03,377 --> 00:34:04,711
too much about Spinoza
in this context,

852
00:34:04,778 --> 00:34:06,513
but I'll just point out
that in his <i>Ethics,</i>

853
00:34:06,580 --> 00:34:08,582
this work that he writes about

854
00:34:08,649 --> 00:34:09,783
how religion
was supposed to work,

855
00:34:09,850 --> 00:34:11,785
he puts it in the form
of a Euclidean proof--

856
00:34:11,852 --> 00:34:14,021
like a very strange thing to do.

857
00:34:14,087 --> 00:34:18,258
But not so strange,
because what he's trying to do

858
00:34:18,325 --> 00:34:20,994
is to generate
certain knowledge, right?

859
00:34:21,061 --> 00:34:22,963
The truths of geometry were
things, you start

860
00:34:23,030 --> 00:34:25,098
from these absolutely
incontrovertible axioms,

861
00:34:25,165 --> 00:34:26,900
and everything that follows
from it cannot be denied.

862
00:34:26,967 --> 00:34:29,303
So, this is what you do if
you're trying to say, like,

863
00:34:29,369 --> 00:34:31,572
"You simply cannot disagree
with my religious views,

864
00:34:31,638 --> 00:34:33,373
try as you might,
because they're in this form."

865
00:34:33,440 --> 00:34:35,375
They're in the form
of, like, axiom, axiom, axiom,

866
00:34:35,442 --> 00:34:36,743
which you accept,

867
00:34:36,810 --> 00:34:40,948
and then, theorem, theorem,
theorem, proof, proof, proof.

868
00:34:41,014 --> 00:34:43,550
I'm gonna slightly,
very rapidly summarize

869
00:34:43,617 --> 00:34:45,385
a wonderful book
by Alma Steingart,

870
00:34:45,452 --> 00:34:48,355
a historian of mathematics...

871
00:34:49,089 --> 00:34:51,091
about this, where...

872
00:34:52,092 --> 00:34:53,727
There's two real visions
of how mathematics works

873
00:34:53,794 --> 00:34:55,629
that are quite incompatible
in a way.

874
00:34:55,696 --> 00:34:59,166
There's the old-timey vision
where we start with some axioms.

875
00:34:59,233 --> 00:35:01,635
This is kind of what we do
when we teach geometry.

876
00:35:01,702 --> 00:35:03,237
We start with some axioms,
and we're like,

877
00:35:03,303 --> 00:35:04,438
"We simply must
agree on these.

878
00:35:04,505 --> 00:35:05,706
"We have no choice.

879
00:35:05,772 --> 00:35:08,008
And then everything that follows
them must be true."

880
00:35:08,075 --> 00:35:10,110
The more modern view
is rather different.

881
00:35:10,177 --> 00:35:11,979
It says the job
of the mathematician

882
00:35:12,045 --> 00:35:14,047
is given some axioms,

883
00:35:15,649 --> 00:35:17,985
figure out what
follows from them.

884
00:35:18,051 --> 00:35:21,755
But we're not saying
that the axioms are true.

885
00:35:21,822 --> 00:35:23,156
You might have one set
of axioms.

886
00:35:23,223 --> 00:35:24,558
I might have another.

887
00:35:24,625 --> 00:35:26,860
I might be doing geometry in
the classical Euclidean sense.

888
00:35:26,927 --> 00:35:29,997
You might be doing some other
kind of non-Euclidean geometry.

889
00:35:30,063 --> 00:35:31,732
And that's okay.

890
00:35:32,599 --> 00:35:35,936
That's a very different
viewpoint.

891
00:35:37,304 --> 00:35:39,039
And what Alma's book is about
is the way

892
00:35:39,106 --> 00:35:41,975
that this point of view
of mathematics becomes dominant,

893
00:35:42,042 --> 00:35:44,111
especially in the United States,

894
00:35:44,178 --> 00:35:46,380
in the great flowering
of American mathematics

895
00:35:46,446 --> 00:35:48,749
in the first half
of the 20th century.

896
00:35:48,815 --> 00:35:50,350
And I wanna kind of make
the claim--

897
00:35:50,417 --> 00:35:51,552
I'm sort of going out beyond

898
00:35:51,618 --> 00:35:52,953
what she says in her book,
right?

899
00:35:53,020 --> 00:35:54,354
I kind of feel strongly
about this.

900
00:35:54,421 --> 00:35:56,356
I wanna make the claim
that it's not an accident,

901
00:35:56,423 --> 00:35:57,558
that here is where it happens.

902
00:35:57,624 --> 00:35:59,359
It is an American mathematics
this point of view

903
00:35:59,426 --> 00:36:00,527
really becomes dominant.

904
00:36:00,594 --> 00:36:01,695
Because you got to remember

905
00:36:01,762 --> 00:36:05,365
that for Spinoza,
writing in the Netherlands,

906
00:36:05,432 --> 00:36:08,101
people are just, like,
straight-up murdering each other

907
00:36:08,168 --> 00:36:09,870
over religious differences
all the time.

908
00:36:09,937 --> 00:36:11,605
Like, that's pretty normal.

909
00:36:11,672 --> 00:36:14,741
And so, for him,
religious difference--

910
00:36:14,808 --> 00:36:16,777
that is a problem
that has to be stamped out.

911
00:36:16,844 --> 00:36:18,979
You have to get people to agree
on what is true,

912
00:36:19,046 --> 00:36:22,983
because if you don't,
they murder each other.

913
00:36:23,050 --> 00:36:28,055
The United States has
a rather different tradition--

914
00:36:28,789 --> 00:36:31,725
what I would call, like, a
tradition of political pluralism

915
00:36:31,792 --> 00:36:34,528
where I think in
an American political context,

916
00:36:34,595 --> 00:36:37,998
we do not demand that difference
of opinion be stamped out.

917
00:36:38,065 --> 00:36:39,199
It's quite the opposite,
actually.

918
00:36:39,266 --> 00:36:41,668
The whole point of the American
system is to say,

919
00:36:41,735 --> 00:36:43,270
can you actually run a country

920
00:36:43,337 --> 00:36:46,039
if people have, like,
radically different beliefs

921
00:36:46,106 --> 00:36:48,041
about what is the case?

922
00:36:48,108 --> 00:36:49,376
And yet, somehow it works

923
00:36:49,443 --> 00:36:53,981
and somehow, the country
trundles along anyway.

924
00:36:54,047 --> 00:36:57,284
So, I think that this popularity
of this view of mathematics

925
00:36:57,351 --> 00:36:58,452
is something like that.

926
00:36:58,519 --> 00:37:00,621
It fits with the United States,

927
00:37:00,687 --> 00:37:03,724
because it says not what are the
things we all have to believe,

928
00:37:03,790 --> 00:37:07,728
but let's think carefully about
how you get to an outcome

929
00:37:07,794 --> 00:37:10,497
from whatever axioms
you happen to start with.

930
00:37:10,564 --> 00:37:12,165
But mine might be different
from yours.

931
00:37:12,232 --> 00:37:15,869
We're not gonna try to stamp out
that difference.

932
00:37:15,936 --> 00:37:18,272
All right, we're coming to near
the end of our time together,

933
00:37:18,338 --> 00:37:21,375
so let me make
a few closing remarks.

934
00:37:23,277 --> 00:37:28,148
We started with this idea of
classification, and I think...

935
00:37:28,215 --> 00:37:29,550
It leads to
all kinds of questions.

936
00:37:29,616 --> 00:37:31,051
If we don't have
this firm boundary

937
00:37:31,118 --> 00:37:32,586
between extroverts
and introverts,

938
00:37:32,653 --> 00:37:36,390
we don't have a firm boundary
between mammals and birds

939
00:37:36,456 --> 00:37:39,026
or even, like,
odd and even numbers,

940
00:37:39,092 --> 00:37:42,262
or even numbers
and things that aren't numbers,

941
00:37:42,329 --> 00:37:44,331
it creates
a certain kind of anxiety.

942
00:37:44,398 --> 00:37:45,732
Remember,
I told you there's a reason

943
00:37:45,799 --> 00:37:48,168
people want this classification,
because we're uncomfortable.

944
00:37:48,235 --> 00:37:50,904
It makes you think maybe there
is no such thing as a mammal

945
00:37:50,971 --> 00:37:52,105
or an extrovert and a number.

946
00:37:52,172 --> 00:37:53,941
I'm gonna reject that.

947
00:37:54,007 --> 00:37:55,943
We all know there's such a thing
as a mammal.

948
00:37:56,009 --> 00:37:57,311
That's not actually a problem.

949
00:37:57,377 --> 00:37:59,913
We all know there's such a thing
as an extrovert; I am one.

950
00:37:59,980 --> 00:38:02,115
We all know that there's
such a thing as a number.

951
00:38:02,182 --> 00:38:03,717
We all teach about them,
we use them.

952
00:38:03,784 --> 00:38:04,885
They are real things.

953
00:38:04,952 --> 00:38:06,086
The fact that they don't have

954
00:38:06,153 --> 00:38:07,487
a hard boundary around them,
I think,

955
00:38:07,554 --> 00:38:09,289
just fundamentally
should not be seen as a problem

956
00:38:09,356 --> 00:38:10,557
for whether these things
actually exist.

957
00:38:10,624 --> 00:38:13,861
We could say more about that,
but not in this talk.

958
00:38:13,927 --> 00:38:17,097
Even here--
let's come back to Shakespeare.

959
00:38:17,164 --> 00:38:21,502
Why is the expression
"Neither fish nor flesh?"

960
00:38:21,568 --> 00:38:23,036
Does anybody know?

961
00:38:23,103 --> 00:38:24,271
Why was this the issue,

962
00:38:24,338 --> 00:38:27,574
is this a fish
or is it a land animal?

963
00:38:29,710 --> 00:38:32,312
Don't let this guy answer again.

964
00:38:32,379 --> 00:38:33,714
Yeah?

965
00:38:33,780 --> 00:38:35,716
- Attendee 2: If I remember
correctly, it's because, like,

966
00:38:35,782 --> 00:38:38,585
anything that lived in the water
was classified as fish,

967
00:38:38,652 --> 00:38:40,287
and then, anything
that was on the land

968
00:38:40,354 --> 00:38:42,890
was classified as animal.
- Yeah, but why do we care?

969
00:38:42,956 --> 00:38:45,092
Why do we care what's a fish
and what's an animal?

970
00:38:45,158 --> 00:38:47,294
- Attendee 3: Does it have to do
with the fact

971
00:38:47,361 --> 00:38:49,196
that, like, fish on Fridays
in the Catholic Church,

972
00:38:49,263 --> 00:38:50,364
during that kind of time,

973
00:38:50,430 --> 00:38:52,499
like, you could have fish
on Fridays, but not eat--

974
00:38:52,566 --> 00:38:54,334
- Exactly,
what can you eat on Lent?

975
00:38:54,401 --> 00:38:56,069
That's the issue.

976
00:38:57,804 --> 00:39:00,674
So, that is why
this is a controversy,

977
00:39:00,741 --> 00:39:04,011
why something like an otter
is controversial.

978
00:39:04,077 --> 00:39:06,246
And I just want to comment

979
00:39:06,313 --> 00:39:10,083
that actually, this is an issue
for the Catholic Church.

980
00:39:10,150 --> 00:39:11,418
And you might say, "Okay, well,

981
00:39:11,485 --> 00:39:13,520
"so the Catholic Church
must have some actual definition

982
00:39:13,587 --> 00:39:16,823
of what's a mammal
and what's a fish."

983
00:39:16,890 --> 00:39:18,091
But no.

984
00:39:18,158 --> 00:39:20,194
Actually, I think the church
understands very well

985
00:39:20,260 --> 00:39:21,695
that this is a fuzzy boundary.

986
00:39:21,762 --> 00:39:24,298
And in particular,
I wrote a little list

987
00:39:24,364 --> 00:39:27,601
that, there are
Catholic dioceses

988
00:39:29,503 --> 00:39:31,839
where you can eat an otter
on Lent

989
00:39:31,905 --> 00:39:33,340
and others where you cannot.

990
00:39:33,407 --> 00:39:36,109
You can eat a muskrat if you
live in the Diocese of Detroit,

991
00:39:36,176 --> 00:39:38,011
a beaver if you live in Quebec,

992
00:39:38,078 --> 00:39:40,414
an alligator
if you live in New Orleans,

993
00:39:40,480 --> 00:39:42,883
and in Venezuela a capybara.

994
00:39:42,950 --> 00:39:46,887
All of those are fish
in those dioceses alone,

995
00:39:46,954 --> 00:39:48,522
according to the
local authorities.

996
00:39:48,589 --> 00:39:51,191
And so, I just want to emphasize
that I think that honestly,

997
00:39:51,258 --> 00:39:54,394
if you work on these things
carefully...

998
00:39:54,461 --> 00:39:55,629
I think you always find

999
00:39:55,696 --> 00:39:57,364
that these boundaries
are not very solid

1000
00:39:57,431 --> 00:39:58,966
and they have to be thought of
contextually.

1001
00:39:59,032 --> 00:40:02,603
So, I am gonna take sort of
one side of this controversy.

1002
00:40:02,669 --> 00:40:04,605
I'm sort of with
the United States of America

1003
00:40:04,671 --> 00:40:06,006
and the Catholic Church on this,

1004
00:40:06,073 --> 00:40:08,475
that, like,
the question about definition

1005
00:40:08,542 --> 00:40:10,344
should not be, like,
"What is this thing?"

1006
00:40:10,410 --> 00:40:13,080
It should be, "What should we
define this thing to be

1007
00:40:13,146 --> 00:40:14,348
according to our purposes?"

1008
00:40:14,414 --> 00:40:16,550
What is an odd number,
and what is an even number?

1009
00:40:16,617 --> 00:40:20,521
It depends what we're trying
to do with those numbers.

1010
00:40:20,587 --> 00:40:23,257
And maybe I'll close with...

1011
00:40:25,926 --> 00:40:28,428
a wonderful book
by Richard Dedekind,

1012
00:40:28,495 --> 00:40:32,699
which addresses this question
about what numbers are.

1013
00:40:32,766 --> 00:40:35,369
This is what I teach
when I teach Math 521 at UW.

1014
00:40:35,435 --> 00:40:37,271
I teach the ideas in this book.

1015
00:40:37,337 --> 00:40:39,072
But it has a fascinating title,

1016
00:40:39,139 --> 00:40:40,574
<i>Was sind und was sollen</i>
<i>die Zahlen?</i>

1017
00:40:40,641 --> 00:40:42,576
It sounds great in German,
not so great in English.

1018
00:40:42,643 --> 00:40:43,744
In English, it says,

1019
00:40:43,810 --> 00:40:47,614
<i>What are Numbers</i>
<i>and What Should They Be?</i>

1020
00:40:47,915 --> 00:40:50,050
So, I wanna close here.

1021
00:40:51,685 --> 00:40:53,086
I think it might be, again,

1022
00:40:53,153 --> 00:40:54,755
surprising to people
outside mathematics

1023
00:40:54,821 --> 00:40:56,356
that a mathematician
would write a book

1024
00:40:56,423 --> 00:40:59,660
with the word "should" in it
at all.

1025
00:41:00,027 --> 00:41:01,128
If you have that view,

1026
00:41:01,195 --> 00:41:03,497
that mathematics is about
what things are.

1027
00:41:03,564 --> 00:41:05,332
But I don't think that's true.

1028
00:41:05,399 --> 00:41:07,267
Mathematics is a human activity,

1029
00:41:07,334 --> 00:41:10,938
and every human activity
has values embedded in it.

1030
00:41:11,004 --> 00:41:13,941
And that's why the word "should"
is appropriate.

1031
00:41:14,007 --> 00:41:17,611
Because when we ask
what is the definition

1032
00:41:17,678 --> 00:41:20,414
of an odd or an even number,
or for that matter when we ask,

1033
00:41:20,480 --> 00:41:23,183
what's the definition
of a mammal or a bird,

1034
00:41:23,250 --> 00:41:25,586
there is a value question there.

1035
00:41:25,652 --> 00:41:29,456
You might have your definition,
and I might have mine.

1036
00:41:29,523 --> 00:41:31,625
But there are actually values
that we share

1037
00:41:31,692 --> 00:41:32,926
where we can argue about

1038
00:41:32,993 --> 00:41:35,128
what is a good definition
and what is a bad definition.

1039
00:41:35,195 --> 00:41:36,296
Not what is <i>the</i> definition.

1040
00:41:36,363 --> 00:41:38,432
That's what I'm trying
to get us past.

1041
00:41:38,498 --> 00:41:41,602
But rather some definitions
actually are better than others.

1042
00:41:41,668 --> 00:41:46,373
And which one is better
might depend on our purposes.

1043
00:41:46,440 --> 00:41:48,642
So, with that,
because time is short,

1044
00:41:48,709 --> 00:41:50,777
I think I'm gonna close.

1045
00:41:50,844 --> 00:41:54,147
I'll just say some of what
I talked about today,

1046
00:41:54,214 --> 00:41:55,949
some of it is in books
I've already written,

1047
00:41:56,016 --> 00:41:58,352
and some of it is in the book
I'm writing right now,

1048
00:41:58,418 --> 00:42:00,954
which will be out in May 2027
if people wanna read about it.

1049
00:42:01,021 --> 00:42:03,156
And now, I wanna leave us
some time to ask questions.

1050
00:42:03,223 --> 00:42:05,058
And so, I have a picture
of myself

1051
00:42:05,125 --> 00:42:09,563
in a suit with question marks
all over it, and a goat.

1052
00:42:09,630 --> 00:42:12,332
So, I'm happy to take questions.

1053
00:42:12,399 --> 00:42:13,867
Yeah.
- Attendee 3: Completely random,

1054
00:42:13,934 --> 00:42:16,503
how do you feel about imaginary
versus real numbers,

1055
00:42:16,570 --> 00:42:18,272
and where does zero belong?

1056
00:42:18,338 --> 00:42:20,908
- So, the question is,
how do I feel--

1057
00:42:20,974 --> 00:42:22,910
And I like that you phrased
the question that way,

1058
00:42:22,976 --> 00:42:24,511
'cause it is about our feelings.

1059
00:42:24,578 --> 00:42:25,913
How do I feel
about the distinction

1060
00:42:25,979 --> 00:42:28,248
between real
and imaginary numbers,

1061
00:42:28,315 --> 00:42:31,752
and where does zero fit
in that question?

1062
00:42:31,818 --> 00:42:36,857
And again, it's a kind of
notation that we're stuck with,

1063
00:42:36,924 --> 00:42:38,458
but it's terrible,
because I like feel like

1064
00:42:38,525 --> 00:42:41,228
the imaginary numbers
are no more imaginary

1065
00:42:41,295 --> 00:42:42,396
than the real number.

1066
00:42:42,462 --> 00:42:43,564
I mean, I feel like

1067
00:42:43,630 --> 00:42:44,965
the imaginary numbers
are not that imaginary

1068
00:42:45,032 --> 00:42:47,301
and the real numbers
are not that real, frankly.

1069
00:42:47,367 --> 00:42:50,137
I would say
they are of equal status.

1070
00:42:50,204 --> 00:42:51,872
But I think...

1071
00:42:55,876 --> 00:42:57,878
But I think, again...

1072
00:43:00,214 --> 00:43:04,985
it's a great example of the
folk definition of a number,

1073
00:43:05,052 --> 00:43:06,620
when you're a small child,

1074
00:43:06,687 --> 00:43:11,859
it's the answer to a question
of the form, "How many?"

1075
00:43:11,925 --> 00:43:14,695
And "I" is definitely not that,
right?

1076
00:43:14,761 --> 00:43:16,830
Whatever it is, it's not that.

1077
00:43:16,897 --> 00:43:19,933
But, then again, like,
neither is the square root of 2,

1078
00:43:20,000 --> 00:43:22,369
which is ostensibly
a real number.

1079
00:43:22,436 --> 00:43:24,671
And, you know, frankly,

1080
00:43:26,173 --> 00:43:28,442
I don't even think a fraction
is really an answer

1081
00:43:28,509 --> 00:43:30,544
to a question
of, like, how many.

1082
00:43:30,611 --> 00:43:33,747
It is not a number
you can count to.

1083
00:43:35,115 --> 00:43:37,251
Maybe that's, like,
a little bit more of a how much

1084
00:43:37,317 --> 00:43:38,418
than a how many.

1085
00:43:38,485 --> 00:43:39,820
So, you know,

1086
00:43:39,887 --> 00:43:41,822
I think this process of, like,
expanding our number system

1087
00:43:41,889 --> 00:43:43,457
is a process of expanding

1088
00:43:43,524 --> 00:43:47,761
what is our definition
of a number bit by bit.

1089
00:43:48,996 --> 00:43:50,931
But I do think we sort of--

1090
00:43:50,998 --> 00:43:54,101
I do think
it's on some level wrong

1091
00:43:54,168 --> 00:43:55,769
to sort of say to our students

1092
00:43:55,836 --> 00:43:59,673
"There is a square root of -1,
and it's called I."

1093
00:43:59,740 --> 00:44:01,675
That's probably
how we mostly do it.

1094
00:44:01,742 --> 00:44:02,843
I think there's something

1095
00:44:02,910 --> 00:44:04,244
a little bit philosophically
shady about that.

1096
00:44:04,311 --> 00:44:06,046
I think we should be honest
about the fact

1097
00:44:06,113 --> 00:44:08,015
that, like, we have decided

1098
00:44:08,081 --> 00:44:10,951
that such a thing
should be called a number.

1099
00:44:11,018 --> 00:44:13,720
Why should it be called
a number?

1100
00:44:13,787 --> 00:44:15,722
I mean, there's a lot of answers
to that question.

1101
00:44:15,789 --> 00:44:17,925
I'm an algebraist, so I kind of
have an algebraic answer.

1102
00:44:17,991 --> 00:44:19,660
I like to call it
the duck principle.

1103
00:44:19,726 --> 00:44:20,827
Do you know this?

1104
00:44:20,894 --> 00:44:22,829
If it quacks like a duck,
then it's a duck.

1105
00:44:22,896 --> 00:44:24,097
So, I feel like

1106
00:44:24,164 --> 00:44:26,099
if you can add them
and multiply them like numbers,

1107
00:44:26,166 --> 00:44:27,701
then they're numbers
as far as I'm concerned.

1108
00:44:27,768 --> 00:44:28,902
That's my duck principle answer.

1109
00:44:28,969 --> 00:44:30,737
But notice I'm using "should,"
right?

1110
00:44:30,804 --> 00:44:32,272
I'm saying there is a reason

1111
00:44:32,339 --> 00:44:34,541
that I've decided
to declare there to be a thing

1112
00:44:34,608 --> 00:44:37,010
which I'm calling
the square root of -1.

1113
00:44:37,077 --> 00:44:40,981
But I don't think we should act
as if it is just there

1114
00:44:41,048 --> 00:44:42,616
and we're informing people
that it's there.

1115
00:44:42,683 --> 00:44:44,017
We should be honest
about the fact

1116
00:44:44,084 --> 00:44:47,754
that we're defining it
into existence.

1117
00:44:48,121 --> 00:44:49,356
Yeah.

1118
00:44:49,423 --> 00:44:51,358
- Attendee 4: So, I feel like
there's an intersection

1119
00:44:51,425 --> 00:44:55,262
between philosophy and math that
you very much pointed out today.

1120
00:44:55,329 --> 00:44:59,132
Are professors also like that
that aren't you,

1121
00:44:59,199 --> 00:45:01,468
or are you unique in that?

1122
00:45:01,535 --> 00:45:03,337
[audience laughs]

1123
00:45:04,137 --> 00:45:06,406
- So, there's a question
about, like,

1124
00:45:06,473 --> 00:45:08,609
to what extent is the way
I'm talking about this,

1125
00:45:08,675 --> 00:45:10,344
like, normal or mainstream?

1126
00:45:10,410 --> 00:45:12,179
And I'm definitely--
exactly as you say.

1127
00:45:12,246 --> 00:45:13,647
So, the questioner pointed out

1128
00:45:13,714 --> 00:45:16,984
that I'm kind of using stuff
from philosophy

1129
00:45:17,050 --> 00:45:18,418
and mixing it
with stuff from math.

1130
00:45:18,485 --> 00:45:21,688
And the answer is that
there is a very long tradition

1131
00:45:21,755 --> 00:45:25,158
of doing exactly that:
mixing philosophy and math.

1132
00:45:25,225 --> 00:45:26,560
If I'm to be honest,
there's people

1133
00:45:26,627 --> 00:45:29,997
who are much more learned
at that intersection than I am.

1134
00:45:30,063 --> 00:45:32,766
And I would say that
the farther back in time you go,

1135
00:45:32,833 --> 00:45:35,802
the less of a disciplinary
distinction there is.

1136
00:45:35,869 --> 00:45:40,374
So, I think most of the really
wonderful early mathematicians,

1137
00:45:40,440 --> 00:45:42,376
you would kind of call them
philosophers of math

1138
00:45:42,442 --> 00:45:44,111
at the same time.

1139
00:45:45,846 --> 00:45:48,382
First of all, I don't think
I used his name in these slides,

1140
00:45:48,448 --> 00:45:50,551
but Hilbert, who is the one
of the great mathematicians

1141
00:45:50,617 --> 00:45:53,487
of the early 20th century
from Germany,

1142
00:45:53,554 --> 00:45:55,189
he is the person
most responsible

1143
00:45:55,255 --> 00:45:57,124
for bringing us
into this kind of

1144
00:45:57,191 --> 00:45:59,960
you get to choose your own
axioms world we live today.

1145
00:46:00,027 --> 00:46:01,161
And I would definitely call him,

1146
00:46:01,228 --> 00:46:02,329
like, a philosopher
of mathematics

1147
00:46:02,396 --> 00:46:04,364
as well as
a working mathematician.

1148
00:46:04,431 --> 00:46:07,668
So, I think this intersection
is, like, real,

1149
00:46:07,734 --> 00:46:10,204
and it's well-populated.

1150
00:46:12,773 --> 00:46:14,508
Okay, so thank you guys
so much for coming.

1151
00:46:14,575 --> 00:46:15,909
I'm still here,
I'm not going anywhere.

1152
00:46:15,976 --> 00:46:18,312
I'm gonna go sit out by the lake
in a little bit

1153
00:46:18,378 --> 00:46:19,479
'cause it's so nice today.

1154
00:46:19,546 --> 00:46:20,681
So, if people have
more questions,

1155
00:46:20,747 --> 00:46:21,882
feel free to come up afterwards.

1156
00:46:21,949 --> 00:46:23,884
I have no schedule,
so thank you guys so much.

1157
00:46:23,951 --> 00:46:25,953
[audience applauds]